Many people add to the content of the Slide Rule Universe, sometimes with pictures, sometimes with research, other times with original technical content or articles. We are fortunate to have this article contributed by Dr. Alan Morris on a topic that will interest many visitors to this site.

The discussion of accuracy and precision is especially relevant to slide rules, because both of the terms are often misused, and frequently misunderstood where slide rules are concerned. Some of the recent discussion threads on the mail list clearly illustrate this. While this article is by no means the final resolution of this issue, it does help clarify the situation, and gives a concrete method of evaluating precision and accuracy in your own rules. Both very useful accomplishments.

One thing to remember is that both the accuracy and precision of any computing device (whether a slide rule, calculator or computer) need to be suitable for the intended purpose. If you are working with components of 5% accuracy, for example, in an electronic design, precision of 0.01% is not really meaningful. For computing complex orbital trajectories, you will no doubt want all the precision and accuracy (not to mention attention to small details like UNITS, as NASA learned recently on the Mars project) you can possibly obtain.

In addition, no matter what precision is required, you have probably ASSUMED that the method used for calculation is always intrinsically accurate, no matter what level of precision it is capable of. This is not true, and after reading this article, you will understand much better why all these things are so.

Introduction
Patently, the most widely published slide rule instruction
manual was the book by Kells, Kern and Bland which
accompanied every K+E Log Log Duplex slide rule ever
manufactured. This book, in its beginning pages, usually
"Section 3," has a paragraph entitled "Accuracy of the Slide
Rule." That this very paragraph discusses slide rule
precision, not slide rule accuracy, shows that a gross error
was propagated for decades by the authors engaged by K+E
to write their manuals and, further, that no editor or stylist
ever corrected the paragraph. Because there is a world of
difference between the meanings of the two words
"accuracy" and "precision" this present paper is written to
clarify the meanings of the contructs of accuracy and
precision, and will do so by means of examples which are
related to scales usage.
This paper will also present a discussion of both initial
manufacture and longevity effects on slide rule accuracy; a
discussion of accuracy and precision characteristics of slide
rules shorter and longer than ten inches as compared with
the accuracy and precision of ten inch rules; and, in an
Appendix, the paper will present a twelve-level slide rule
accuracy evaluation sequence for 10" LogLog Duplex slide
rules, a sequence developed through an extensive program
of slide rule evaluations.
The Laboratory Scale Experiment
In an Elementary Physics course laboratory, a class of 30
students, arranged in teams of two, are given the following
materials and assignment:
Each team is handed a 12" steel scale, a piece of white
bond paper, a sharpened pencil, and a 10x magnifying
glass; the steel scales are engine divided in 1/100 inch
intervals.
The student teams are instructed to make two marks an
arbitrary distance apart of their sheets of paper.
The student teams are then instructed to use their steel
scales to measure the distance between their two marks 100
times, using the magnifying glass,
recording each measurement made in tabular form on their
data sheet, and to alternate measuring between the two
team members. The distance measurements are to be
made to the nearest 1/100 of an inch, with interpolations to
be made should the pencil mark lie between two adjacent
l/100 inch graduations of the steel measuring scale.
When the series of 100 measurements is complete, the
student teams are to compute the average and the standard
deviation of their individual sets of data, and then hand in
their results.
When, at the following session of the Elementary Physics
laboratory, the student teams are handed back their
measurement results papers from the previous laboratory
session, the students find that each team's results, i.e., the
average and the standard deviation computed, are graded
with a big red "F" for Fail. Naturally, the students want to
know why their very careful work had universally been
graded Fail.
To answer the students' questions, the teacher handed back
to each student team one of the steel scales that they had
used at the previous laboratory session. The students
were directed to study the fine writing at the left end of the
rules, which writing stated:
"Linear shrink: steel, puddled, l:50"
The teacher explained that, although the scales looked like
fine, accurate, engraved steel rules, in fact these steel rules,
marked "12", were actually 12.24" long! In other words, a l2"
measurement made with the rule would be about a l/4"
longer than 12", that a 6" measurement made with the rule
would be about an l/8" longer than 6", and so forth. The
teacher further explained that these rules were used to size
patterns for sand molding of puddled steel alloy, and that
castings made with that alloy shrink one part in 50 in every
direction upon cooling. Thus the pattern created for the
casting using this alloy would correspondingly have to be
made one part in 50 larger in every direction in order to
assure that the cooled casting would be of the desired
dimensions.
Then to drive home hard the pivotal point of the entire
exercise, the teacher lectured the students thus:
Use of the shrink rules to measure distances in actual inches
and fractions of inches, down to l/l00" and, further, down to
an estimated l/l000 of an inch by interpolation, was totally
erroneous, since the shrink rules could be counted on to
make measurements, say, of a 12" distance to only ¬" scale
intervals, not l/100" or, more ridiculously, to l/l000" by
interpolative estimates.
Thus the shrink rules were precise, because measurements
made with the rules could be determined to within 1/l00 inch,
and interpolations could be made to an estimated l/l000 inch.
But for making true measurements, e.g., the measured
distance between the marks made with sharpened pencil
during the experiment, the rules were not accurate.
In conclusion, the teacher stated, the rules appeared to be
accurately made, but the rules were not accurate for
measuring actual distances, the rules were only precise.
Other Accuracy vs. Precision Examples
Having presented the Laboratory Scale Experiment findings,
the following examples will serve to further demonstrate the
total, and absolute, difference in the meanings of the two
distinct constructs: accuracy, and precision:
A gas tank gage in an automobile has a finely divided scale
which can be used to read to the nearest 1/10 gallon.
However, unbeknownst to the operator of the vehicle, a
miscreant has secretly bent the needle of the gage at a point
near the needle's pivot, a point that is hidden by the fascia of
the instrument panel. The miscreant who bent the gage
needle arranged the bend so that when the needle showed
the gas tank as being "Full," the tank would actually be half-
full. The gage then becomes an instrument that is precise,
but that is woefully inaccurate.
A watch dial is graduated in l/5th of a second intervals
between each minute mark. Thus the watch is precise. But
unbeknownst to the person using the watch to observe the
time, the watch is five minutes slow; reading a time to the
nearest 1/5th second with this watch, while being precise, is
ridiculous, because the time reading is five whole minutes
away from the true time - the watch is inaccurate.
The Constructs of Accuracy and Precision as Applied to
Log Log Duplex Rules
Contrary, then, to what Kells, Kern and Bland stated in every
edition of the K+E instruction book, the readings the authors
describe relate only to precision, i.e., the scale intervals that
permit a user to read or set the rule to three or more places.
Having the scale properties of precision states nothing
about, and has no relationship whatever to, the properties of
accuracy of the rule.
The accuracy of a slide rule has, at the time of manufacture,
everything to do with how the engraving or printing of all of
the scale graduations correspond with the true
mathematically-computed positions of every graduation on
the rule.
Assuming then for the moment that a particular slide rule
was accurately laid down at the time of manufacture, nothing
specific can be said about the effects on that rule's accuracy
down through time; those effects can include not only
damage and abuse, but also in the case of a wood or paper
rule, shrinkage or expansion, non-uniformly in a single
direction or differentially in numerous directions throughout
the entire volume of the rule.
In the case of the Log Log Duplex rule there is of course the
all-important consideration of transfer of calculations from
front to rear and from rear to front sides of the rule. Thus, in
a Duplex rule the accuracy-damaging effects of time are
potentially greatly enhanced because of the two-sided
referencing that must be done with that style of rule, even if it
is assumed that the Duplex rule was laid down accurately,
both sides, and both sides in registry, at the time of
manufacture.
Additional Accuracy-Limiting Factors in LogLog Duplex
Rules
When a rule is manufactured, the wood may not have been
properly aged, and so the body or slide or both may warp,
either in a single curve, or in a wavy curve, or the slide
portion may warp differently than the body portions. In the
latter two cases, the slide at various points along the mating
edges, will lie either above or below the adjacent surface of
the body, leading to parallax errors on reading and on
setting, even if all of the rule's graduations were accurately
laid down during manufacture. A rule can also become
curved, one wave, multiple waves, differential waves,
through bad storage or careless handling, or from warping
that occurs over time; the limiting parallax effects above-
described also apply under these circumstances.
Either at the time of manufacture, or through aging, some of
the body or slide edges may lose planar flatness, and flare
out at some or all points along the body mating edges, or at
the slide edges, or at all four mating edges, body and slide.
This flaring-of-edges effect introduces parallax errors on
reading and on setting.
It may prove impossible to bring front and rear cursor
hairlines into perfect coincidence while at the same time
bringing the pair of hairlines into perfect registration with the
front and rear sides of the rule. The usual import of this
impossibility, should it arise, is that the front and rear sides
were either not in registration at the time of manufacture
or the front and rear sides through time have proceeded out
of overall registration.
Another accuracy-limiting cursor effect is related to the fit of
the cursor to the slide rule body. Even if the cursor hairlines
are in perfect coincidence front-to-rear, if the cursor is
slightly loose on the body in the transverse direction, having
some slack in that direction, then it is possible that:
a. The cursor can become angled with respect to the
surface of the body, causing inaccurate readings from front-
to-rear because the front-to-rear axis of the hairlines is not
perpendicular to the body of the rule
b. The cursor may shift position when the rule is
flipped over to utilize the other side of the rule body.
c. The cursor may lie fully flat on one side of the
rule, causing the hairline of the cursor on the other side of
the rule to be too high above the surface of the other side of
the rule, leading to parallax errors in accuracy of reading and
setting.
The cursor cannot be so tightly fitted on the body of the rule
so as to be capable of being moved only with difficulty, yet
the optimum free play of the cursor in the transverse position
can be afforded with only a few thousandth's of a inch of
transverse movement. However, having this near-perfect fit
of the cursor means that any dirt that gets under the cursor
windows must be removed; this removal of dirt can be
accomplished easily by use of triangularly shaped, slightly
moistened, slips of 20 lb. white paper, where the tip of
the paper triangle is introduced under the cursor window,
and then the cursor is slid back and forth atop the wider
portions of the paper triangle.
There must be minimum gap widths between the mating
scale edges, for if these gaps are too wide, there will be
accuracy errors on reading and setting the rule. Gap width
can be a function of maladjustment of the adjustable stator,
but due to possible differential shrinkage and expansion of a
rule through time, it may be impossible to reduce the gap by
binding down with the adjustable slider without locking the
slider in place.
Some have suggested to this writer that a slide rule might
expand or contract along its length direction in such a way
that it, if originally accurately laid down, will remain
accurate. Leaving out Pickett metal rules, K+E, Dietzgen
and Hemmi rules are all made of wood. Hemmi rules are
made of a superior and more stable wood, bamboo, than the
wood, mahogany, of which K+E and Dietzgen rules are
made. Wood is an non-homogenous material and there is
no reason why wood, on expansion or on contraction, would
do so in an absolutely linear and uniform manner. If such
were indeed possible, the rule at every point along its length,
body and slide, both sides, would have to expand or contract
with a uniformity of l/l000 of an inch, a certain impossibility.
To make realistic, but at the same time totally impracticable,
the linear expansion and/or contraction suggestion, the rule
would have to be constructed from heavy bars of platinum-
iridium alloy, an alloy having an exceedingly low coefficient
of linear expansion or contraction. Until October 1960, the
international meter was defined as the distance between
two marks on a platinum-iridium bar housed in Paris. To
present an idea of the level of accuracy involved with the
standard of length, in October 1960, by international
agreement, the meter was redefined to be l,650,763.73
wavelengths in vacuo of the orange-red spectral line of
krypton 86.
Another effect that contributes to the potential masking of
rule inaccuracies lies in the thickness of the graduations.
Pickett rules in general have thicker graduations than
Hemmi, K+E and Dietzgen rules. When the writer has
conducted evaluations of Pickett rules, many have been
noted to be inaccurately printed, but some Pickett rules that
have been found to be accurate do meet the higher Levels
(see Appendix) of accuracy through the agency of
somewhat too-thick graduation lines. The writer has
observed that the graduations of 1945 Hemmi rules are
thinner than the graduations found on the last-produced
1975 Hemmi rules.
Visual Acuity and the Slide Rule
The resolving power of the human eye is related to the
visual angle subtended by the finest detail that the eye
can distinguish. However, it is a property of the human
eye, a property long made use of in optical devices such
as split-image rangefinders, that the eye can distinguish
line objects and the coincidence of or the lack of
coincidence of line objects, at visual angles far less than
those at the limit of the resolving power of the eye. For
example, one can easily see a distant telephone or power
line although the visual angle subtended by the distant
line object is much smaller than the visual angle
subtended at the resolving power limit of the eye. One
can distinguish at a distance if one line is close to, but not
touching, another line.
The line-distinguishing property of the human eye makes
facile the reading and setting of a slide rule, since the eye
can work well with critical line alignment or line non-
alignment; these being the visual tasks involved with slide
rule calculations.
Slide Rules Shorter Than 10"
"Pocket" slide rules of the LogLog Duplex design have 5"
scale lengths. Even if such a rule is accurately laid down,
there are two effects which serve to limit the accuracy
of a 5" rule as compared with a l0" rule:
a. The thickness of the graduation lines on the 5"
rule cannot be less than the thickness of the graduation
lines found on the 10" rule, while, logically, the
graduations on a 5" rule should be « the thickness of the
graduations on a l0" rule. As discussed above in this
paper, too-thick graduation lines serve to mask
inaccuracies, and thus lead to errors on setting and on
reading.
b. The 5" rule is less precise than a l0" rule, since
the 5" rule is not as finely divided as a 10" rule,
necessarily so, as otherwise, the 5" rule's scales would
be rendered useless through overcrowding.
Slide Rules Longer Than 10"
Examples of rules longer than 10" include the 20" Log
Log Duplex rule, certain cylindrical rules, large circular
rules, classroom wall demonstration rules, and the
multiply-staved Thacher rule. It should be clear from this
paper that rules longer than 10" can certainly provide
more precision of setting and of reading, but again it
will here be reemphasized that the scale properties of
precision has no relationship whatever to the properties of
accuracy of the rule.
If, for example, on a 10" rule, 1.01 and 9.95 can each be
set on a graduation line, and if, for example, on 20" rule,
1.005 and 9.975 can each be set on a graduation line,
there is absolutely no warranty that the increase of
precision afforded by the 20" rule due to the increased
fineness of the graduations on the 20" rule will result in
more accurate calculations with the 20" rule than can be
made with the 10" rule. This is because longer rules
made of wood or paper could not be manufactured with
greater accuracy than a 10" rule, and rules longer than
10" cannot withstand the effects of longevity, namely,
differential expansion and contraction, warping, edge-
flaring, as well as can a 10" rule. The costs of
manufacturing an accurate rule longer than 10" would far
exceed the costs of manufacturing an accurate 10" rule.
It is well known in one of the scale procedures of highest
accuracy, the field of ruling the lines of diffraction
gratings, that the lengthy engraving machine lead screw
is the single most extraordinarily costly element of the
entire machine. Correspondingly, if a manufacturer set
out to make accurate slide rules longer than 10", he
doubtless would not utilize a wooden base for the slide
rule body; also, his engraving machines would
have to be crafted to be accurate over a distance of a
least twice the length of a l0" rule, radically increasing the
costs of the machines and correspondingly the sales
prices of long accurate rules.
All this is not to say that all 10" rules were all accurately
made, in fact, there is no proof that very many 10" slide
rules at all were accurately made. What we do have in
the way of proof, however, is the converse proof, made
inferentially, through this writer's exhaustive and
continuing program of slide rule evaluations, being as
follows:
The fact that the writer has identified several near perfect
and several fully perfectly accurate 10" Log Log Duplex
slide rules, albeit many, many years having passed since
these identified slide rules were manufactured, means
that at one time or another the machinery for producing
the 10" scale length duplex slide rule was capable of
producing a totally accurate rule.
Obviously, it is only through unknown circumstances that
a slide rule that originally was accurately manufactured
would present itself today as still being an accurate slide
rule.
Definitions
For scale readings and/or settings, as obtain in using
slide rules, then:
a. To how many places can the scale be read or
set is the measure of precision of the scale.
b. How close a reading or setting is to the true
value of the number is the measure of the accuracy of the
scale.
In Conclusion
The 10" Log Log Duplex slide rule is the optimum design for
a slide rule, for that design is the single slide rule design that
at the same time provides:
a. Potential for overall accuracy.
b. Two-sided design that allows for openness of
scales layout, front and rear, and that makes complex
calculations involving trigonometric, log, ln, reciprocal,
square root, cube root, and exponential functions easy to
accomplish, all in a compact package that can be easily
grasped and that balances well in the hand.
As the single function of a slide rule is to enable the user to
make accurate calculations, and as accurate calculations
can only be made using a slide rule that is accurate, then it
is this writer's opinion that an accurate 10" Log Log Duplex
slide rule is the premier slide rule and that any slide rule that
is less than accurate, or is of any design or configuration
other than 10" Log Duplex, is merely a curiosity.
Appendix: Accuracy Evaluation Sequence For 10" Log
Log Duplex Slide Rules
This accuracy evaluation presented below increases greatly
in difficulty as a candidate rule is sequenced through each
succeeding Level.
Tools Required
8X optical loupe
Set of screwdrivers of proper tip width and tip sharpness
Wooden or plastic small mallet for tapping slider and
adjustable stator (can be smooth wooden handle of old-style
small screwdriver)
Preliminary Evaluation
Warping - examine rule, end-on, for warping; if excessive
(see present paper), reject the rule.
Flaring - examine mating edges of scales to note the extent
of any flaring; if excessive (see present paper), reject the
rule.
Cursor Fit - examine fit of cursor; if loose transversely (see
present paper) reject the rule.
Slider/Stator Gaps - check the gap widths between mating
edges (see present paper); if gaps cannot be uniformly
minimized, reject the rule.
Level Rating Evaluation
Level l: Check C vs. D on front side.
Level 2: Check A vs. B on rear side.
Level 3: Check C and D vs. A and B.
Level 4: Check CF vs. DF.
Level 5: Check D vs. Sin.
Level 6: Check CF and DF vs. D and Sin.
Setting A: Align stators to slider.
Level 7: Check C vs. D vs. CF vs. DF.
Level 8: Check A vs. B vs. Sin vs. D.
Level 9: Check C vs. D vs. CF vs. DF vs. A vs. B vs. Sin vs.
D.
Cleaning: Clean the slide rule following the methods of
Bruce Babcock (Oughtred Journal, Vol. 2, No. 2, October
1993, p. 18); clean the underside of the cursor windows
(see present paper).
Setting B: Align cursor hairline to body on front side.
Level 10: Check all front side scales, bottom to top of
rule.
Setting C: Align cursor hairline to body on rear side.
Level 11: Check all rear side scales, bottom to top of
rule.
Setting D: Align cursor to bring front side hairline into
coincidence with rear side hairline, while at the same time
taking care to maintain the front side alignment of Setting B
as well as the rear side alignment of Setting C.
Level 12: Check all scales, front side to rear side,
bottom to top of rule.